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Questions tagged [copula]

A copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the unit interval.

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What is the probability of $\alpha$- quantile

I didn't understand why the $P(X \leq q_{\alpha}^- (X))=\alpha$ for $0<\alpha<1$. Please help me with this. The setup for it: X is a real random variable and the lower $\alpha$-quantile defined ...
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Variable change in Copula for the joint pdf of correlated random variables

Let $f_{X,Y}(x,y)$ be the joint probability density of correlated random variables $X$ and $Y$ based on a Copula $C$ (Gaussian in my case) where $f_X(x)$ and $f_Y(y)$ are the marginal probability ...
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Expected value of copula density function

I have a copula density function $c(u, v)$ (I can assume that $c(u, v)$ is continuous). I want to prove that there is an upper bound $M$ for the following expected value: $$ \mathbb{E} (|\log c(U_1, ...
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Joint probability distribution of the min and max of two successive rolls of a die.

I was going through the article coping with copulas. The following situation was considered there. A die is rolled twice. $X_1$ is the random variable representing the minimum of the two rolls, $X_2$ ...
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Find the copula to minimize KL-divergence [closed]

Consider the set of all N-dimensional copulas denoted by $\mathcal{C}^N$. Notice that each copula $C$ is a joint distribution on the state space $[0,1]^N$ with uniform marginal distributions. Now ...
copula's user avatar
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What assumptions/properties, if any, characterize the Gaussian copula

I would like to know what assumptions and/or properties characterize the Gaussian copula. I am interested in the case of arbitrary dimensionality, and am not so interested in properties specific to ...
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Endogeneity Analysis without the access of raw data?

I currently have the correlation/covariance matrix for a set of variables and the results of regression analysis but lack access to the raw dataset. Under these constraints, is it feasible to conduct ...
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Multivariate sampling for inherently positive data with given mean values and covariance matrix

Context : We perform measurements of a set of parameters $(p_1,p_2,...)$ to retrieve the experimental mean of each parameters and the covariance matrix between them. The means of the parameters are ...
SoleGoodman's user avatar
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Any copula to force one variable larger than the other in the joint?

I want to transform two known marginal distributions into a joint distribution by using copula. As I understand commonly used copulas cannot make sure that in the joint distribution one variable is ...
Nicolas Fabre's user avatar
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Does Sklar's Theorem work when $P(X_k \leq t)$ are not continuous?

Wikipedia here describes for $X = (X_1,...,X_n): \Omega \rightarrow \mathbb{R}^n$ with each $F_k(t) = P(X_k \leq t)$ continuous, there is a pushforward measure induced on $[0,1]^n$ via $(F_1(X_1),...,...
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Vine Copulas: Conditional Copula cdf in higher dimensions

Hello :) I am currently working on a project where I am stuck calculating the following term: $u_{i(e)|D(e)} = C_{i(e)|D(e)}(u_{i(e)}|D(e))$ $u_{j(e)|D(e)} = C_{i(e)|D(e)}(j_{i(e)}|D(e))$ The ...
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Dependence between stochastic processes

Suppose $(I_t)_{t \in \mathbb{R}^+}$ and $(D_t)_{t \in \mathbb{R}^+}$ and $(C_t)_{t \in \mathbb{R}^+}$ are three dependent Poisson processes with parameters $\lambda_I$, $\lambda_D$, and $\lambda_C$ ...
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Why copula is a cdf?

I am doing an assignment on copula functions. In my assignment, copula function is defined as follows: Def A copula function is simply a multivariate uniform distribution over $[0, 1]$ hypercube. For ...
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Calculation on Quadrant Probability for Bivariate data using Copula

I have a question regarding the computation on bivariate probability when using copula function. Let $u=F_X(x)$ and $v=F_T(t)$ be the CDF for marginals of $X$ and $T$ respectively, a joint ...
AHMAD FAIZ BIN MOHD AZHAR MSC2's user avatar
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Implied Correlation from Gaussian Copula

I am building a spreadsheet model that allows marginal distributions to be correlated together using a Gaussian copula (with prescribed correlation matrix). The inputs into the model are the ...
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Indefinite integral | How to compute the second-order partial derivatives of Mixed (2 gaussian & 2 binary) Gaussian Copula?

This is a problem about "Computing the Mixture Gaussian Copula with 2 normal (continuous) variables and 2 binary (discrete) variables". Problem background I have two continuous r.v. and two ...
黄琳恩's user avatar
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How is the copula $C\big(X_1, 1/X_1, \exp(-X_1)\big)$ equal to the copula $C(X_1, - X_1, -X_1)$ and what is its form?

Let $X_1$ be a positive random variable with a continuous cumulative distribution function and $C$ the copula of $(X_1, 1/X_1, \exp(-X_1))$. Why is $C$ also the copula of $(X_1, -X_1, -X_1)$ and what ...
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Generate uniform random numbers with a certain dependency structure

Let's assume I wanted to sample from a $\mathcal N(\boldsymbol 0, \boldsymbol\Sigma)$ distribution, where $$\boldsymbol\Sigma = \begin{pmatrix} 1 & \rho \\ \rho & 1\end{pmatrix}$$ for some $\...
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Stochastic ordering of dependent random variables given their sum and a common copula

I am interested in pointers for the stochastic ordering of continuous dependent variables with common copulas. I have a vector $X=(X_1,X_2,...,X_N)$ and a vector $Y=(Y_1, Y_2, ..., Y_N)$. X and Y ...
GM Williams's user avatar
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Does uniform distribution on every square $[0,a]^2$ along diagonal imply uniform CDF on the entire $ [0,1]^2$

Let $F: [0,1]^2\to R $ be a continuous cdf with uniform marginals, i.e., $F(x,1)=x$ and $F(1,y)=y$. Suppose $F$ is symmetric, i.e., $F(x,y)=F(y,x)$. Suppose we also know that $F(a,a)=a^2$ for all $a\...
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How to prove that the multivariate Frechet-Hoeffding lower bound is not copula when $n \geq 3$?

I want to show that the multivariate Frechet-Hoeffding lower bound given by $C^-(u_1,\dots, u_n) = \left(\sum_{i=1}^{n}{u_i} - (n-1)\right)_+$ is not a copula when $n \geq 3$. My question is: Is it ...
booster's user avatar
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Semi-survival copula

I'm currently doing some reading on copula function. For a bivariate cdf, by Sklar's theorem, there exists a copula function,C, such that $P(U<u,V<v) = C(u, v)$. As written by Nelsen (2006), A ...
AHMAD FAIZ BIN MOHD AZHAR MSC2's user avatar
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Multivariate t distribution: Find probability of region enclosed by constant-density hypersurface

I am working with a multivariate t distribution, say of dimension p. Given a point P = (x1, ..., xp) in the sample space I need to calculate the probability of the region of the sample space enclosed ...
Andrew Kirk's user avatar
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Monotonicity of Parametric Bivariate Copula w.r.t. $\theta$

Let $C(u_1,u_2;\theta)$ be a bivariate parametric copula. I know if $C(u_1,u_2;\theta)$ is the Gaussian copula, then $\partial_{\theta} C(u_1,u_2;\theta)>0$ for any $u_1$ and $u_2$ (it follows from ...
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How do I derive a pair-copula decomposition for a joint density function?

In Section 4.1 of Analyzing Dependent Data with Vine Copulas (Czado), the author decomposes a three-dimensional joint density function into bivariate copula densities and marginal density functions. I’...
Leon's user avatar
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Why can this joint distribution function be written as this integral of a conditional distribution function?

In Section 3.8 of Dependence Modeling with Copulas (Joe), the author starts with the following. In this section, we show how Sklar's theorem applies to a set of univariate conditional distributions, ...
Leon's user avatar
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Finding a value $\alpha$ such that a function $c$ is a copula density of $(U_1, U_2)$

Let $$c(u_1,u_2) = 1 + \alpha(1- 2u_1)(2- 2u_2)$$ where $u_1, u_2 \in (0,1)$. The question is, for which $\alpha$ is the function $c$ a copula density for $(U_1, U_2)$. So my idea was to compute the ...
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Beyond vine copulas

I know that vine copulas allow for different dependency models between pairs of variables. I wondered if they are also used to model dependency models between subsets (beyond pairs) of variables or if ...
Jeremy's user avatar
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Sum of dependent random variables and copulas

I have two dependent continuous random variables (RVs) $X$ and $Y$ and I'm interested in determining the CDF of the sum, i.e., $F_{X+Y}(t) = \mathbb{P}(X+Y \leq t)$. I know the marginal of $X$ and $Y$ ...
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Proving that copulas are Lipschitz continuous

A copula is a function $C:[0,1]^2\to[0,1]$ such that $C(x,0)=C(0,x)=0$ for all $x\in[0,1]$, $C(x,1)=C(1,x)=x$ for all $x\in[0,1]$, and \begin{equation}\label{ineq} C(x_2,y_2)-C(x_1,y_2)-C(x_2,y_1)+C(...
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Calculating the joint cumulative distribution function from a junction tree

Assume I have the following Junction Tree between random variables $X_1,\dots,X_7$ that exactly describes the sets variables with non-zero Mutual Information (Alternatively it's the last tree in a ...
Daniel P's user avatar
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Product of correlated random variables and its transformation

There is an interesting result, saying that if $Z_1, Z_2$ are standard normal random variables with a correlation $\rho\in (-1,1)$, then the product $Z=Z_1Z_2$ has a density function explicitly given ...
Albert Paradek's user avatar
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Combining conditional probabilities using an unconditional copula

First: just a bit of background on copulas. Suppose we have a pair of continuous random variables $Y_1, Y_2$ with distribution functions $F_1(y_1)=P(Y_1\leq y_1)$ and $F_2(y_2)=P(Y_2\leq y_2)$. Let ...
Hudson Cooper's user avatar
2 votes
1 answer
283 views

Proving that $C(x_1,x_2)=\max\{x_1+x_2-1,0\}$ is a copula

I have the following problem. I need to prove the copula property for the function $C(x_1,x_2)=\max\{x_1+x_2-1,0\}$, better known as the lower Fréchet-Hoeffding-Boundary. A copula is defined as a ...
Dani's user avatar
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Frequently used methods to estimate copula of discrete random variable in high dimension

I am new to the estimation of copula with discrete marginal distributions. For example, I have the data of 30 discrete random count variables from $X_1$ to $X_{30}$, some $X_j$ has Poisson ...
InTheSearchForKnowledge's user avatar
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1 answer
264 views

Kendall’s tau between $X$ and $1/Y$

I'm currently studying for my statistics exam by doing exercises from the book Statistics and Data Analysis for Financial Engineering with R examples. I'm struggling with this exercise from chapter 8: ...
MrVandebilt's user avatar
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1 answer
200 views

Proof that Pearson correlation is/isn't a functional of the copula for a given pair of random variables

I have a growing interest and respect for the subject of copulas initially thanks to comments made on stats.SE by kjetil-b-halvorsen. The most interesting to me right now is the following: "[T]...
Galen's user avatar
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Sampling from Gaussian Copula with a conditional distribution approach

In the paper "Cheng and al. (2007)" on pages 193-194, the authors propose an algorithm that generates variables with a given copula function $C$ being the joint distribution. This procedure ...
Louis De Montalte's user avatar
2 votes
1 answer
513 views

Showing that the lower bivariate Fréchet-Hoeffding bound is a copula.

I want to show that the bivariate Fréchet-Hoefdding lower bound is indeed a copula. The bivariate function is defined by: $W(x,y)=\max\{x+y-1,0\}, \ x,y\in[0,1] $ Definition "Copula": A two-...
mathstruggler's user avatar
1 vote
1 answer
486 views

How to deal with copulas?

I want to model a couple of variables $(X,Y)$ using copulas. My idea is to model the marginal distributions and then use copulas to combine marginal distributions and copula to get a joint ...
jocelinbordet's user avatar
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Kendall's tau for archimedian copula

Consider a 2-dimensional archimedian copula with generator $\psi$ and $\gamma:=\psi^{-1}$, i.e. $$C(u,v)=\gamma(\psi(u)+\psi(v))$$ I want to show that $$\tau=1-\int_{0}^{\infty}t\gamma'(t)^2dt$$ I ...
user826130's user avatar
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Find $\lim_{x\uparrow 1}\mathbb{P}(F_Y(Y)>x|F_X(X)>x)$ where $X,Y$ poisson

Let $X=Y_1+N,Y=Y_2+N$ where $N\sim \text{pois}(\lambda), Y_1\sim \text{pois}(\lambda_1), Y_2\sim \text{pois}(\lambda_2)$ and $N,Y_1,Y_2$ are independent. I'm asked to find $$\lim_{x\uparrow 1}\mathbb{...
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$P(u_1\leq\alpha,u_2\leq\beta)$ and $P(u_1>\alpha,u_2>\beta)$ and copulas

I know this is meant to be elementary, but I don't know why I cannot convince myself -- given a copula $C(u_1,u_2)=P(u_1\leq \alpha,u_2\leq\beta)$, why is $P(u_1>\alpha,u_2>\beta)=1-\alpha-\beta-...
user107224's user avatar
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Verifying if a function is 2-increasing

In the book An Introduction to Copulas (2006), by Roger B. Nelsen, on page 8, the author defines a function $H$ as 2-increasing if \begin{equation} V_H(B)=H(x_2,y_2)-H(x_2,y_1)- H(x_1,y_2)+H(x_1,y_1)\...
Louis De Montalte's user avatar
1 vote
1 answer
234 views

How can I get an intuition about Copula?

I am really struggling with getting an intuition about copulas. I have red many articles and I am stuck at what is the concept/idea behind it. For example if I have two random variables X and Y and I ...
Alessia Comes's user avatar
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117 views

Correlation matrix of Gaussian copula

The question is related to the Gaussian copula. Let $\Phi(x)$ denote the cdf of standard Normal distribution. Let $(X_1, X_2) \sim \mathcal{N}(0,\Sigma)$ be joint Normal with covariance matrix $$ \...
zxzx179's user avatar
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Joint Distribution of Two Dependent Variables having the Marginals

From two Independent Normal Random Variables: $X \sim N(\mu_1,\sigma) $ and $Y\sim N(\mu_1,\sigma)$, I created two DEPENDENT random variables $Z$ and $W$: $Z= X - Y$ $W= X - g(Y)$ where $g(\...
Joan's user avatar
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How to compare the entropy of copulas?

Three different bivariate copulas are shown below with increasing degrees of dependence (parameter $\theta$). Differential entropy is a measure of disorder in a probability density like the copula. ...
develarist's user avatar
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1 vote
1 answer
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Impact of copula choice on quantiles (sum of random variables)

I am trying to get my head around the impact of different dependence structures (copulas) on the risk (quantiles) of a sum of dependent random variables (with arbitrary marginals). In a multivariate ...
noclue's user avatar
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1 answer
170 views

Why is copula entropy negative?

Most statistical measures are non-negative. Shannon entropy, $H(X)$, a statistical measure of disorder or uncertainty in probability distributions, is also non-negative, despite it having a negative ...
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